Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere: The Exceptional Case

Johann S. Brauchart (), Peter D. Dragnev (), Edward B. Saff () and Robert S. Womersley ()
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Johann S. Brauchart: Graz University of Technology, Institute of Analysis and Number Theory
Peter D. Dragnev: Indiana University - Purdue University, Department of Mathematical Sciences
Edward B. Saff: Vanderbilt University, Center for Constructive Approximation, Department of Mathematics
Robert S. Womersley: University of New South Wales, School of Mathematics and Statistics

A chapter in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 2018, pp 179-203 from Springer

Abstract: Abstract We consider the minimal discrete and continuous energy problems on the unit sphere 𝕊 d $$\mathbb {S}^d$$ in the Euclidean space ℝ d + 1 $$\mathbb {R}^{d+1}$$ in the presence of an external field due to finitely many localized charge distributions on 𝕊 d $$\mathbb {S}^d$$ , where the energy arises from the Riesz potential 1∕r s (r is the Euclidean distance) for the critical Riesz parameter s = d − 2 if d ≥ 3 and the logarithmic potential log ( 1 ∕ r ) $$\log (1/r)$$ if d = 2. Individually, a localized charge distribution is either a point charge or assumed to be rotationally symmetric. The extremal measure solving the continuous external field problem for weak fields is shown to be the uniform measure on the sphere but restricted to the exterior of spherical caps surrounding the localized charge distributions. The radii are determined by the relative strengths of the generating charges. Furthermore, we show that the minimal energy points solving the related discrete external field problem are confined to this support. For d − 2 ≤ s

Date: 2018
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DOI: 10.1007/978-3-319-72456-0_10

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